Algebra Qualifying Examination
May, 2005
Directions:
1. Answer all questions. (Total possible is 100 points.)
2. Start each question on a new sheet of paper.
3. Write only on one side of each sheet of paper.
Policy on Misprints:
The Qualifying Exam Committee tries to proofread the exams as carefully as possible. Nevertheless, the exam may contain a few misprints. If you are convinced a
problem has been stated incorrectly, indicate your interpretation in writing your
answer. In such cases, do not interpret the problem in such a way that it becomes
trivial.
Notes: All rings are unitary. All modules are unitary. Q is the rationals, R the reals, C
the complexes, Z the integers, and Zn is the integers modulo n (or Z/nZ).
1. (10 points) Let V be a finite dimensional vector space over a field F . Let T : V → V
be a linear transformation of rank 1. Prove that T is either diagonalizable or nilpotent
but not both.
2. (10 points) Let G = {a1 , a2 , ..., an } be an abelian group of order n. What are the possibilities for a1 +a2 +· · ·+an and when do each occur? Your work should demonstrate
that your answer is correct.
3. (10 points) Let G = Sn , the symmetric group on n letters. Then G acts on the set
X = {1, 2, ..., n} in a natural way, namely, σ · i = σ(i). For σ ∈ S n let
Fix(σ) = |{i ∈ X : σ(i) = i}|. So Fix(σ) is the number of elements of X fixed by σ.
Determine the integer
Σσ∈Sn Fix(σ)
and prove your answer.
(over)
4. (20 points) Let R = Z[i], the ring of Gaussian integers.
a. Let p ∈ Z be a prime integer. Show p is prime in R if x2 + y 2 = p has no integral
solutions. (Hint: Remember the norm function N : R → Z.)
b. Show that 11 does not divide 4n2 + 1 for all n ∈ Z.
5. (20 points) Let R be a commutative ring.
a. Prove that, for any R-module M , M ⊗R R ∼
= M.
b. Compute Q ⊗Z Z2 . Your work should prove that your answer is correct.
6. (10 points) Let f (x) = x4 + 1 ∈ Z[x].
2
a. Show that, for all odd primes p, f (x) divides xp − x in Z[x].
b. Show that f (x) is reducible when considered as a polynomial in Z p [x] for each
prime p.
7. (5 points) Suppose F is an extension field of a field K. Suppose a ∈ F is algebraic
such that [K[a] : K] is odd. Show that K(a) = K(a2 ) ⊆ F .
8. (15 points) Let x be an indeterminate. Your work on the following should demonstrate
that your answer is correct.
a. Find [Q(x) : Q(x4 )].
b. Find |AutQ(x4 ) Q(x)|.
c. Show whether or not Q(x) is Galois over Q(x4 ).